Trivial measures are not so trivial

TL;DR

This paper explores the structure of trivial computable measures, which are supported on countable sets, revealing complexities previously overlooked in the study of algorithmic randomness.

Contribution

It uncovers the richer structure of trivial computable measures and their role in algorithmic randomness, expanding understanding beyond non-uniform measures.

Findings

Trivial measures have more complex structure than previously thought.

Algorithmic randomness behaves differently under trivial measures.

The study broadens the scope of randomness analysis to include trivial measures.

Abstract

Although algorithmic randomness with respect to various non-uniform computable measures is well-studied, little attention has been paid to algorithmic randomness with respect to computable \emph{trivial} measures, where a measure $\mu$ on $2^\omega$ is trivial if the support of $\mu$ consists of a countable collection of sequences. In this article, it is shown that there is much more structure to trivial computable measures than has been previously suspected.